{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:TUONWYIOZZRUDDQUG3BMRTCACF","short_pith_number":"pith:TUONWYIO","canonical_record":{"source":{"id":"2605.21998","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.SP","submitted_at":"2026-05-21T04:56:01Z","cross_cats_sorted":[],"title_canon_sha256":"0f317392c15ab6f4ff6be3c00871023ba1643171e0a9f90b76ef0979c8233080","abstract_canon_sha256":"f7f02d5af4c8bf58121487111b3ef5f63f537f681568c377837f0ab045e17e7d"},"schema_version":"1.0"},"canonical_sha256":"9d1cdb610ece63418e1436c2c8cc4011447a831e5c2820a646163fcc1d2d95cd","source":{"kind":"arxiv","id":"2605.21998","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.21998","created_at":"2026-05-22T01:04:19Z"},{"alias_kind":"arxiv_version","alias_value":"2605.21998v1","created_at":"2026-05-22T01:04:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.21998","created_at":"2026-05-22T01:04:19Z"},{"alias_kind":"pith_short_12","alias_value":"TUONWYIOZZRU","created_at":"2026-05-22T01:04:19Z"},{"alias_kind":"pith_short_16","alias_value":"TUONWYIOZZRUDDQU","created_at":"2026-05-22T01:04:19Z"},{"alias_kind":"pith_short_8","alias_value":"TUONWYIO","created_at":"2026-05-22T01:04:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:TUONWYIOZZRUDDQUG3BMRTCACF","target":"record","payload":{"canonical_record":{"source":{"id":"2605.21998","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.SP","submitted_at":"2026-05-21T04:56:01Z","cross_cats_sorted":[],"title_canon_sha256":"0f317392c15ab6f4ff6be3c00871023ba1643171e0a9f90b76ef0979c8233080","abstract_canon_sha256":"f7f02d5af4c8bf58121487111b3ef5f63f537f681568c377837f0ab045e17e7d"},"schema_version":"1.0"},"canonical_sha256":"9d1cdb610ece63418e1436c2c8cc4011447a831e5c2820a646163fcc1d2d95cd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-22T01:04:19.752239Z","signature_b64":"ZR3nRdbafpGy+wSfsZuNCsUQFGXKrMEEnwPjcF7qttU+f2vW6SDwzZ1BOb2sS4Gg0cT8D23hEy25OBsMXuSJAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9d1cdb610ece63418e1436c2c8cc4011447a831e5c2820a646163fcc1d2d95cd","last_reissued_at":"2026-05-22T01:04:19.751509Z","signature_status":"signed_v1","first_computed_at":"2026-05-22T01:04:19.751509Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.21998","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-22T01:04:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KbmVnQtw//qvMiJdxgXqROhR4Ipb1qexjizs1UNqzrdmnFL4cOHk112PuqxThKrKtUY9b7oDzm8yxO2NDHs8Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T18:54:46.894394Z"},"content_sha256":"c07ee48098a9cbf584be759faeb8cbc28a463e8347902529cb4fc888bb8a5da8","schema_version":"1.0","event_id":"sha256:c07ee48098a9cbf584be759faeb8cbc28a463e8347902529cb4fc888bb8a5da8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:TUONWYIOZZRUDDQUG3BMRTCACF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Minimally $(k,k)$-edge-connected graphs via spectral radius","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Dan Li, Huiqiu Lin, Yu Wang","submitted_at":"2026-05-21T04:56:01Z","abstract_excerpt":"For $l > 1$, the $l$-edge-connectivity $\\kappa'_l(G)$ of a connected graph $G$ is defined as the minimum number of edges whose removal leaves a graph with at least $l$ components. A graph is minimally $(k,l)$-edge-connected if $\\kappa'_l(G)\\geq k$ but for any edge $e\\in E(G)$ satisfies that $\\kappa'_l(G-e)< k$. Motivated by two foundational extremal problems: Brualdi and Solheid's problem [SIAM J. Algebra Discrete Methods (1986)] for graphs of fixed order: determine sharp upper bounds for the spectral radius over graph families and characterize extremal graphs; and its fixed size analogue prop"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.21998","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.21998/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-22T01:04:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7vILGtyrDDSQkR+akMs6whF/GIHtWI+nQea/MAu+NsQipaG4fFYBOJdKZ7AUrSBVx3q7gn7oCr5M079UMUYFBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T18:54:46.895171Z"},"content_sha256":"9335c1c0e77de8e259440cd06b6489e38f456c0015717b514d22ccc787489e91","schema_version":"1.0","event_id":"sha256:9335c1c0e77de8e259440cd06b6489e38f456c0015717b514d22ccc787489e91"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TUONWYIOZZRUDDQUG3BMRTCACF/bundle.json","state_url":"https://pith.science/pith/TUONWYIOZZRUDDQUG3BMRTCACF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TUONWYIOZZRUDDQUG3BMRTCACF/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-06T18:54:46Z","links":{"resolver":"https://pith.science/pith/TUONWYIOZZRUDDQUG3BMRTCACF","bundle":"https://pith.science/pith/TUONWYIOZZRUDDQUG3BMRTCACF/bundle.json","state":"https://pith.science/pith/TUONWYIOZZRUDDQUG3BMRTCACF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TUONWYIOZZRUDDQUG3BMRTCACF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:TUONWYIOZZRUDDQUG3BMRTCACF","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"f7f02d5af4c8bf58121487111b3ef5f63f537f681568c377837f0ab045e17e7d","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.SP","submitted_at":"2026-05-21T04:56:01Z","title_canon_sha256":"0f317392c15ab6f4ff6be3c00871023ba1643171e0a9f90b76ef0979c8233080"},"schema_version":"1.0","source":{"id":"2605.21998","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.21998","created_at":"2026-05-22T01:04:19Z"},{"alias_kind":"arxiv_version","alias_value":"2605.21998v1","created_at":"2026-05-22T01:04:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.21998","created_at":"2026-05-22T01:04:19Z"},{"alias_kind":"pith_short_12","alias_value":"TUONWYIOZZRU","created_at":"2026-05-22T01:04:19Z"},{"alias_kind":"pith_short_16","alias_value":"TUONWYIOZZRUDDQU","created_at":"2026-05-22T01:04:19Z"},{"alias_kind":"pith_short_8","alias_value":"TUONWYIO","created_at":"2026-05-22T01:04:19Z"}],"graph_snapshots":[{"event_id":"sha256:9335c1c0e77de8e259440cd06b6489e38f456c0015717b514d22ccc787489e91","target":"graph","created_at":"2026-05-22T01:04:19Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2605.21998/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For $l > 1$, the $l$-edge-connectivity $\\kappa'_l(G)$ of a connected graph $G$ is defined as the minimum number of edges whose removal leaves a graph with at least $l$ components. A graph is minimally $(k,l)$-edge-connected if $\\kappa'_l(G)\\geq k$ but for any edge $e\\in E(G)$ satisfies that $\\kappa'_l(G-e)< k$. Motivated by two foundational extremal problems: Brualdi and Solheid's problem [SIAM J. Algebra Discrete Methods (1986)] for graphs of fixed order: determine sharp upper bounds for the spectral radius over graph families and characterize extremal graphs; and its fixed size analogue prop","authors_text":"Dan Li, Huiqiu Lin, Yu Wang","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.SP","submitted_at":"2026-05-21T04:56:01Z","title":"Minimally $(k,k)$-edge-connected graphs via spectral radius"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.21998","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:c07ee48098a9cbf584be759faeb8cbc28a463e8347902529cb4fc888bb8a5da8","target":"record","created_at":"2026-05-22T01:04:19Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"f7f02d5af4c8bf58121487111b3ef5f63f537f681568c377837f0ab045e17e7d","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.SP","submitted_at":"2026-05-21T04:56:01Z","title_canon_sha256":"0f317392c15ab6f4ff6be3c00871023ba1643171e0a9f90b76ef0979c8233080"},"schema_version":"1.0","source":{"id":"2605.21998","kind":"arxiv","version":1}},"canonical_sha256":"9d1cdb610ece63418e1436c2c8cc4011447a831e5c2820a646163fcc1d2d95cd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9d1cdb610ece63418e1436c2c8cc4011447a831e5c2820a646163fcc1d2d95cd","first_computed_at":"2026-05-22T01:04:19.751509Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-22T01:04:19.751509Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZR3nRdbafpGy+wSfsZuNCsUQFGXKrMEEnwPjcF7qttU+f2vW6SDwzZ1BOb2sS4Gg0cT8D23hEy25OBsMXuSJAQ==","signature_status":"signed_v1","signed_at":"2026-05-22T01:04:19.752239Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.21998","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c07ee48098a9cbf584be759faeb8cbc28a463e8347902529cb4fc888bb8a5da8","sha256:9335c1c0e77de8e259440cd06b6489e38f456c0015717b514d22ccc787489e91"],"state_sha256":"163675523a9fcaa2c659130478902bff5a7671a8be437c113bb29deb3187cf62"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"w22eGpvO21qhgDw0N+V8Cl+/edKHVZKHkyeVLQZthvVaTvWcUxBOewtTgZEkiAhTaRsRiIY2ksld/rNcLpI9Cg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T18:54:46.899035Z","bundle_sha256":"45cf97457dcf55d794273a99973237c1d58b584dbfb45fbbac240bb57e866a01"}}